Why absolute value in derivative of inverse cosecant (csc^-1x)

Why absolute value in derivative of inverse cosecant? In this video, you will learn why the derivative of inverse cosecant has an absolute value? Why absolute value in derivative of inverse cosecant? You will also learn “how to find the derivative of cosecant inverse of x”. Negative derivative of cosecant inverse x. Why is differentiation of cosec inverse x always negative? Why is derivative of cosec inverse x always negative? Why is derivative of inverse cosecant always negative? Why is the derivative of arccsc x always negative? Why is differentiation of cosec inverse x containing mod x? Why is derivative of cosec inverse x having mod x? Why is derivative of inverse cosecant having mod x? Why is the derivative of arccsc x has mod x? Why is differentiation of cosec inverse x containing absolute value of x? Why is derivative of cosec inverse x having absolute value of x? Why is derivative of inverse cosecant having absolute value of x? Why is the derivative of arccsc x has absolute value of x? Derivative of inverse cosecant x equals negative 1 over product of x and square root of x squared minus 1. The denominator of right-hand side of derivative contains two terms; one is x and another is square root of x squared minus 1. The sign of the derivative will depend upon sign of x, because the quantity under square root is never negative. It means when x is negative the derivative or slope is positive and vice-versa. But wait, this is not 100% true. When we analyse graph of cosecant inverse of x, the function is not defined for values of x between negative one and positive one. The slope of the function is not defined at 1 and negative 1. Also, when x is greater than 1; slope of this function is negative. Slope will remain negative when x is less than negative 1. So, to make the derivative always negative; we put absolute value sign. This is the reason why the derivative of inverse cosecant has an absolute value.
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